Meromorphic function

A Meromorphic Function on an open subset of  C {\displaystyle \mathbb {C} } is, in the field of Complex Analysis, a function that is holomorphic except at Isolated singularity. These poles must be isolated. The set of all meromorphic functions on a subdomain  U {\displaystyle U} of the complex plane has an advantage over the set of holomorphic functions: it forms not only a ring but also a field. In fact, it can be shown that it is the field of fractions of the ring of holomorphic functions.

## Definition

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Let  U ⊆ C {\displaystyle U\subseteq \mathbb {C} } be open. A meromorphic function  f {\displaystyle f} on  U {\displaystyle U} is a function with a discrete set of singularities  S ⊆ U {\displaystyle S\subseteq U} with

1. f : U ∖ S -> C {\displaystyle f\colon U\setminus S\to \mathbb {C} } is holomorphic.
2. f {\displaystyle f} has a pole at each point  s ∈ S {\displaystyle s\in S} .

We say that  f {\displaystyle f} is meromorphic on  U {\displaystyle U} and write  f ∈ M ( U ) {\displaystyle f\in {\mathcal {M}}(U)} .

Note that a meromorphic function on  U {\displaystyle U} is not defined on the entire set  U {\displaystyle U} , but only on the complement of a discrete subset.

## Remark

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In the definition of singularities, the following three types of singularities were mentioned:

* Removable singularities,
* Poles (of order  m {\displaystyle m} ),
* Essential singularities.

Meromorphic functions may only have poles in the set of singularities  S {\displaystyle S} ; they must not have essential singularities.

## Properties

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1. The sum, difference, and product of two functions  f , g ∈ M ( U ) {\displaystyle f,g\in {\mathcal {M}}(U)} are again meromorphic, so  M ( U ) {\displaystyle {\mathcal {M}}(U)} is an algebra over the ring of holomorphic functions on  U {\displaystyle U} .Let  S {\displaystyle S} be the set of poles of  f {\displaystyle f} and  T {\displaystyle T} the set of poles of  g {\displaystyle g} . Then  S ∪ T {\displaystyle S\cup T} is a discrete subset of  U {\displaystyle U} , and  f + g , f − g , f g {\displaystyle f+g,f-g,fg} are holomorphic on  U ∖ ( S ∪ T ) {\displaystyle U\setminus (S\cup T)} , with removable singularities or poles on  S ∪ T {\displaystyle S\cup T} .
2. If  U {\displaystyle U} is connected,  f , g ∈ M ( U ) {\displaystyle f,g\in {\mathcal {M}}(U)} , and  g ≠ 0 {\displaystyle g\neq 0} , then  f / g {\displaystyle f/g} is meromorphic on  U {\displaystyle U} . In this case,  M ( U ) {\displaystyle {\mathcal {M}}(U)} is a field.Let  S {\displaystyle S} be the set of poles of  f {\displaystyle f} and  T {\displaystyle T} the set of poles of  g {\displaystyle g} . Since  U ∖ T {\displaystyle U\setminus T} is a domain, the zero set  N {\displaystyle N} of  g {\displaystyle g} is a discrete subset of  U {\displaystyle U} , by the Identity Theorem. Now  f / g : U ∖ ( S ∪ T ∪ N ) -> C {\displaystyle f/g\colon U\setminus (S\cup T\cup N)\to \mathbb {C} } is holomorphic, on  S ∪ T ∪ N {\displaystyle S\cup T\cup N} has  f / g {\displaystyle f/g} removable singularities or poles.
3. Locally, every meromorphic function is a quotient of two holomorphic functions. That is, if  f ∈ M ( U ) {\displaystyle f\in {\mathcal {M}}(U)} and  z 0 ∈ U {\displaystyle z_{0}\in U} , there exist a neighborhood  V {\displaystyle V} of  z 0 {\displaystyle z_{0}} and holomorphic functions  p , q : V -> C {\displaystyle p,q\colon V\to \mathbb {C} } such that  f = p / q {\displaystyle f=p/q} .A deeper result shows that for domains  U {\displaystyle U} , such a representation is globally always possible. In this case, the field of meromorphic functions is the field of fractions of the ring of holomorphic functions on  U {\displaystyle U} .

## Equivalent Description as Holomorphic Functions with Values in  C ^ {\displaystyle {\hat {\mathbb {C} }}}

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Another way to describe meromorphic functions on an open set  U ⊆ C {\displaystyle U\subseteq \mathbb {C} } is to define them as holomorphic functions with values in the Riemann sphere.

### Definition

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Let  U ⊆ C {\displaystyle U\subseteq \mathbb {C} } . A function  f : U -> C ^ {\displaystyle f\colon U\to {\hat {\mathbb {C} }}} is called "holomorphic" at a point  z 0 ∈ U {\displaystyle z_{0}\in U} with  f ( z 0 ) ≠ ∞ {\displaystyle f(z_{0})\neq \infty } if there exists a neighborhood  V {\displaystyle V} of  z 0 {\displaystyle z_{0}} such that  f ( V ) ⊆ C {\displaystyle f(V)\subseteq \mathbb {C} } and  f | V : V -> C {\displaystyle f|_{V}\colon V\to \mathbb {C} } is holomorphic at  z 0 {\displaystyle z_{0}} . f {\displaystyle f} is called "holomorphic" at a point  z 0 {\displaystyle z_{0}} with  f ( z 0 ) = ∞ {\displaystyle f(z_{0})=\infty } if  1 f {\displaystyle {\frac {1}{f}}} is holomorphic at  z 0 {\displaystyle z_{0}} in the above sense.

### Poles and Points at Infinity

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Let  f : U -> C ^ {\displaystyle f\colon U\to {\hat {\mathbb {C} }}} be holomorphic with  f ( z 0 ) = ∞ {\displaystyle f(z_{0})=\infty } . If  f {\displaystyle f} is not constant on any neighborhood of  z 0 {\displaystyle z_{0}} , then by the Identity Theorem, there exists a neighborhood  V {\displaystyle V} of  z 0 {\displaystyle z_{0}} such that  f | V ∖ z 0 : V ∖ z 0 -> C {\displaystyle f|_{V\setminus {z_{0}}}\colon V\setminus {z_{0}}\to \mathbb {C} } . Since  f {\displaystyle f} is holomorphic at  z 0 {\displaystyle z_{0}} ,  1 f {\displaystyle {\frac {1}{f}}} has a power series expansion at  z 0 {\displaystyle z_{0}} , say

1 f ( z ) = ∑ k = 0 ∞ a k ( z − z 0 ) k {\displaystyle {\frac {1}{f(z)}}=\sum _{k=0}^{\infty }a_{k}(z-z_{0})^{k}}

Let  m := min { k : a k ≠ 0 } {\displaystyle m:=\min\\{k:a_{k}\neq 0\\}} . Then

1 f ( z ) = ( z − z 0 ) m ∑ k = 0 ∞ a k + m ( z − z 0 ) k ⏟ g ( z ) := {\displaystyle {\frac {1}{f(z)}}=(z-z_{0})^{m}\underbrace {\sum _{k=0}^{\infty }a_{k+m}(z-z_{0})^{k}} _{g(z):=}}

Here, g {\displaystyle g} is holomorphic with  g ( z 0 ) ≠ 0 {\displaystyle g(z_{0})\neq 0} , so  1 / g {\displaystyle 1/g} is holomorphic at  z 0 {\displaystyle z_{0}} . It follows that

f ( z ) ( z − z 0 ) m = 1 g ( z ) , z ∈ V ∖ { z 0 } {\displaystyle f(z)(z-z_{0})^{m}={\frac {1}{g(z)}},\qquad z\in V\setminus \\{z_{0}\\}}

thus,  f {\displaystyle f} in  z 0 {\displaystyle z_{0}} has a pole of order  m {\displaystyle m} .

### Characterization of Meromorphic Functions

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Since poles can be described as points at infinity, we have: A meromorphic function  f {\displaystyle f} on  U ⊆ C {\displaystyle U\subseteq \mathbb {C} } is a holomorphic function  f : U -> C ^ {\displaystyle f\colon U\to {\hat {\mathbb {C} }}} whose points at infinity do not accumulate (equivalently,  f {\displaystyle f} is not constantly equal to  ∞ {\displaystyle \infty } on any component of  U {\displaystyle U} ).

## See also

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* Complex Analysis

### Translation and Version Control

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This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

* Source: Meromorphe Funktion \- URL:

https://de.wikiversity.org/wiki/Meromorphe_Funktion

* Date: 01/07/2024